An arch is one of the oldest and most efficient structural forms in engineering. Whether designing a masonry bridge, a decorative stone entrance, or a reinforced concrete vault, every project begins with the same challenge: accurately deriving the geometric relationships between span, rise, and radius and translating those dimensions into material quantities.
This calculator solves that problem in seconds. Given any two of the three primary dimensions — span ($W$), rise ($H$), and radius ($R$) — it computes the missing third value along with the intrados arc length, extrados arc length, central angle, cross-sectional area of the arch ring, total volume, and estimated mass. The result is a complete geometric and material profile that would otherwise require multiple manual computation steps prone to rounding errors and sign mistakes.
Required Design Parameters
To generate a full arch profile, the following values are needed:
- Span ($W$) — the clear horizontal distance between the two springing points (supports), measured in meters.
- Rise ($H$) — the vertical distance from the springing line (chord) to the highest point of the intrados (inner curve), measured in meters.
- Radius ($R$) — the radius of the circular arc that defines the intrados curve, measured in meters. Required only when span and rise are not both provided.
- Arch Thickness ($T$) — the radial depth of the arch ring, i.e., the distance from the intrados surface to the extrados surface, measured in meters.
- Arch Depth ($D_p$) — the longitudinal dimension of the arch perpendicular to its plane (into the wall or along the bridge), measured in meters.
- Material Density ($\rho$) — the unit weight of the arch material, expressed in kg/m³, used to convert volume into mass.
The first three parameters define the geometry. The last three define the section and material properties necessary for volume and mass estimation.
Theoretical Foundation and Governing Formulas
Relationship Between Span, Rise, and Radius
A circular arch is a portion of a circle. The span $W$ corresponds to the chord of the arc, the rise $H$ corresponds to the sagitta (the maximum perpendicular distance from the chord to the arc), and $R$ is the radius of the generating circle.
These three quantities are not independent. Any two uniquely determine the third through the geometry of the circular segment.
When span and rise are known, the radius is derived from the sagitta-chord-radius identity:
$$R = \frac{H}{2} + \frac{W^2}{8H}$$
This classical formula follows directly from the Pythagorean theorem applied to the right triangle formed by the radius, half the chord, and the apothem ($R - H$).
When span and radius are known, the rise is:
$$H = R - \sqrt{R^2 - \left(\frac{W}{2}\right)^2}$$
This assumes a minor arc (i.e., the rise does not exceed the radius). The constraint $W \leq 2R$ must hold; otherwise no valid circular arc exists for the given pair.
When rise and radius are known, the span is:
$$W = 2\sqrt{2HR - H^2}$$
Again, the constraint $H \leq 2R$ applies. Values of $H > R$ indicate a horseshoe (or ultra-semicircular) arch where the arc exceeds 180°.
Central Angle
The central angle $\theta$ (in radians) subtended by the intrados arc at the center of the generating circle is:
$$\theta = 2 \arcsin\left(\frac{W}{2R}\right)$$
For standard segmental and semi-circular arches where $H \leq R$, this formula yields $\theta \leq \pi$. When $H > R$ (horseshoe arches), the central angle is adjusted to:
$$\theta = 2\pi - 2\arcsin\left(\frac{W}{2R}\right)$$
A perfectly semi-circular arch has $\theta = \pi$ (180°), which occurs when $H = R$ and $W = 2R$.
Arc Length
The intrados arc length $L$ is the product of the radius and the central angle:
$$L = R \cdot \theta$$
The extrados arc length $L_{ext}$ uses the outer radius $R_{ext} = R + T$:
$$L_{ext} = (R + T) \cdot \theta$$
These lengths are critical for estimating formwork, membrane quantities, and reinforcement bar cut-lengths.
Apothem
The apothem $d$ is the perpendicular distance from the center of the generating circle to the chord (springing line):
$$d = R - H$$
For a semi-circular arch, $d = 0$ (the center sits on the chord). For segmental arches $d > 0$, and for horseshoe arches $d < 0$ (the center lies above the chord).
Area of the Circular Segment
The area enclosed between the intrados arc and the springing line — often needed for clearance or waterway calculations — is given by the standard circular segment formula:
$$A_{seg} = \frac{1}{2} R^2 (\theta - \sin\theta)$$
Cross-Sectional Area of the Arch Ring
The arch ring occupies the area between the intrados arc (radius $R$) and the extrados arc (radius $R + T$). This is a circular annulus sector:
$$A_{ring} = \frac{\theta}{2}\left[(R + T)^2 - R^2\right]$$
Which simplifies to:
$$A_{ring} = \frac{\theta}{2}\left(2RT + T^2\right)$$
Volume and Mass
The total solid volume of the arch is found by multiplying the ring area by the longitudinal depth:
$$V = A_{ring} \times D_p$$
The estimated total mass uses the specified material density:
$$m = V \times \rho$$
These are geometric volumes only — they do not account for mortar joints, voids, or spandrel fill. For detailed material take-offs, appropriate waste factors and joint ratios should be applied.
Technical Specifications and Reference Data
The following table provides typical density values for common arch construction materials. These values serve as a starting reference for the density parameter $\rho$.
| Material | Density (kg/m³) | Typical Application |
|---|---|---|
| Normal-weight Concrete (C25–C40) | 2,300 – 2,500 | Cast-in-situ or precast arches |
| Lightweight Concrete | 1,400 – 1,800 | Decorative or non-structural arches |
| Solid Clay Brick | 1,900 – 2,100 | Traditional masonry arch bridges |
| Engineering Brick (Class A/B) | 2,100 – 2,300 | High-strength masonry arches |
| Limestone | 2,200 – 2,600 | Heritage and restoration work |
| Sandstone | 2,000 – 2,500 | Monumental and ecclesiastical arches |
| Granite | 2,600 – 2,800 | Heavy-duty bridge and tunnel arches |
| Natural Stone (average) | 2,400 – 2,700 | General masonry construction |
| Steel (structural) | 7,850 | Steel arch ribs and frames |
| Timber (hardwood, air-dry) | 600 – 900 | Laminated timber arches |
The following table classifies arch types by their rise-to-span ratio $H/W$, a key geometric indicator:
| Rise-to-Span Ratio ($H/W$) | Classification | Structural Character |
|---|---|---|
| < 0.15 | Flat (Jack) Arch | High horizontal thrust; requires robust abutments |
| 0.15 – 0.49 | Segmental Arch | Most common; moderate thrust; efficient material use |
| 0.50 | Semi-circular Arch | Equal span and diameter; classical form; balanced thrust |
| > 0.50 | Horseshoe / Pointed Arch | Central angle exceeds 180°; used in Islamic and Gothic architecture |
Engineering Analysis and Real-World Application
How Rise-to-Span Ratio Governs Structural Behavior
The ratio $H/W$ is the single most informative dimensionless parameter in arch design. A low ratio (flat arch) produces very high horizontal thrusts at the springings, demanding massive abutments or tie rods. As $H/W$ approaches 0.50 (the semi-circular condition), the horizontal thrust decreases and the load path becomes more purely compressive.
Engineers should note that for a fixed span $W$, increasing the rise $H$ simultaneously decreases the radius $R$ when computed via the sagitta formula. This inverse coupling means that a taller arch has a tighter curvature and a shorter radius — a relationship that is counterintuitive to some practitioners.
Thickness-to-Radius Ratio and Stability
The arch thickness $T$ relative to the radius $R$ directly affects structural stability. Jacques Heyman's classical analysis of masonry arches established that the line of thrust must remain within the arch ring for equilibrium. A thicker ring provides a wider corridor for the thrust line, increasing the geometric factor of safety.
A common preliminary rule for unreinforced masonry arches is $T/R \geq 0.04$ to $0.06$, though detailed analysis per EN 1996-1-1 (Eurocode 6) or equivalent standards is always required for final design. The calculator's simultaneous display of both $T$ and $R$ enables rapid checking of this ratio during the conceptual phase.
Interpreting Arc Length for Material Estimation
The intrados arc length $L$ determines the formwork length and the net face area of the arch soffit. The extrados arc length $L_{ext}$ governs the outer surface area, relevant for waterproofing membranes and drainage layers on bridge arches.
The difference $(L_{ext} - L) = T \cdot \theta$ represents the additional length of the outer curve due to the arch thickness. For thick arches or large central angles, this difference becomes significant and must not be neglected in quantity surveying.
Volume-to-Mass Conversion in Practice
The calculated volume $V$ represents the net solid volume of the arch ring. In real masonry construction, mortar joints can constitute 5–15% of the total wall volume depending on unit size and joint width. The calculated mass should therefore be treated as a lower-bound estimate for masonry arches. For concrete arches, the value is more directly applicable.
Frequently Asked Questions
A horseshoe arch is identified when the rise-to-span ratio exceeds 0.50, meaning the rise $H$ is greater than half the span. Geometrically, this occurs when $H > R$, causing the center of the generating circle to fall below the springing line (negative apothem) and the central angle $\theta$ to exceed $\pi$ radians (180°).
In structural terms, horseshoe arches develop thrust vectors that angle inward below the springing line. This creates inward-directed horizontal forces at the supports, which is the opposite of the outward thrust in segmental and semi-circular arches. Abutment design must account for this reversal. Historically, horseshoe arches are prevalent in Moorish and Islamic architecture, where they serve both aesthetic and structural purposes.
Select the mode that matches the two dimensions you can physically measure or that are fixed by project constraints. In most new construction projects, the span is dictated by the opening width and the rise is set by the required clearance height — making Span & Rise the most common starting point.
However, in heritage restoration work, you may have the radius from existing curvature surveys but no clean measurement of the rise due to deformation. In that case, Span & Radius is the appropriate mode. Rise & Radius is useful when designing an arch to match a specific curvature (e.g., aligning with an adjacent existing arch of known radius) while the span is free to vary.
This calculator models single-center circular arches only. Pointed arches (Gothic or lancet) are formed by two arcs struck from different centers, and four-centered (Tudor) arches use four different radii. These compound forms cannot be described by a single radius $R$.
However, each individual arc segment of a pointed or multi-centered arch is a circular arc. You can use this calculator to analyze each segment independently by entering the appropriate partial span and rise for that segment. The arc lengths, areas, and volumes of each segment can then be summed to approximate the full compound arch profile. For precise compound arch analysis, dedicated multi-center arch geometry tools are recommended.
Professional Conclusion
Accurate geometric analysis is the foundation of every arch design — from a small decorative opening in a garden wall to a major masonry bridge span. Manual computation of arc lengths, annular sector areas, and material volumes involves multiple trigonometric and algebraic steps where a single substitution error can cascade through all downstream quantities.
Automated calculation eliminates this risk entirely. By encoding the exact formulas from circular segment geometry and coupling them with section and material parameters, this tool produces a verified, internally consistent set of results in a fraction of the time required for hand calculation. The simultaneous visualization of the arch profile further reduces the chance of input errors by providing immediate geometric feedback.
For professional applications, these results should always be validated against the governing design codes — particularly EN 1996 (Eurocode 6) for masonry structures — and supplemented with structural analysis for thrust lines, stability, and load capacity. The geometric estimation provided here is the essential first step in that process.